https://file.aiquickdraw.com/mathbot/file/172783937364140kgmvxm.jpg

Select the correct answer below:

x=−1 with multiplicity 1; x=−1 with multiplicity 2; and x=−5 with multiplicity 1

x=1 with multiplicity 1; x=−1 with multiplicity 1; and x=−5 with multiplicity 2

x=1 with multiplicity 1; x=−1 with multiplicity 2; and x=−5 with multiplicity 1

x=−1 with multiplicity 2; x=−1 with multiplicity 1; and x=−5 with multiplicity 1

x=1 with multiplicity 2; x=−1 with multiplicity 1; and x=−5 with multiplicity 1

x=1 with multiplicity 2; x=1 with multiplicity 1; and x=−5 with multiplicity 1


To solve the problem, we can analyze the graph of the degree 4 polynomial and identify the zeros and their multiplicities. 

### Step 1: Identify the zeros
The graph crosses or touches the x-axis at specific points, which represent the polynomial's zeros. 

From the image:
- The graph touches the x-axis at \( x = -5 \), indicating a zero with even multiplicity.
- The graph touches the x-axis at \( x = -1 \), but it does not cross, also indicating a zero with even multiplicity.
- The graph crosses the x-axis at \( x = 1 \), indicating a zero with odd multiplicity.

### Step 2: Determine the multiplicities
- **\( x = -5 \)**: Since the graph only touches the x-axis at this point and does not cross, it has **even multiplicity**, likely 2.
- **\( x = -1 \)**: The graph also touches the x-axis here without crossing, so it has **even multiplicity**, likely 2.
- **\( x = 1 \)**: The graph crosses the x-axis, indicating **odd multiplicity**, likely 1.

### Step 3: Select the correct answer
Based on the analysis, the zeros and their multiplicities are:
- \( x = -5 \) with multiplicity 1,
- \( x = -1 \) with multiplicity 2,
- \( x = 1 \) with multiplicity 1.

The correct choice is:
- **\( x = 1 \) with multiplicity 1; \( x = -1 \) with multiplicity 2; and \( x = -5 \) with multiplicity 1.**

Would you like more details on this explanation?

### 5 Related Questions:
1. What happens to the graph when a zero has an odd vs. even multiplicity?
2. How do you determine the degree of a polynomial from its graph?
3. What is the significance of the degree in polynomial behavior?
4. Can a polynomial function have complex zeros, and how would they appear on a graph?
5. How do turning points relate to the degree of a polynomial?

### Tip:
When analyzing polynomial graphs, remember that zeros with even multiplicity touch the x-axis without crossing it, while zeros with odd multiplicity cross the x-axis.

Given the graph of the following degree 4 polynomial function, find all of the zeros and their multiplicities.

Learn how to find the zeros and their multiplicities of a degree 4 polynomial function using its graph. This problem demonstrates the behavior of polynomial functions with zeros of even and odd multiplicities. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation